3.1083 \(\int \frac{(1+x)^{3/2}}{(1-x)^{7/2}} \, dx\)

Optimal. Leaf size=20 \[ \frac{(x+1)^{5/2}}{5 (1-x)^{5/2}} \]

[Out]

(1 + x)^(5/2)/(5*(1 - x)^(5/2))

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Rubi [A]  time = 0.0016671, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {37} \[ \frac{(x+1)^{5/2}}{5 (1-x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)^(3/2)/(1 - x)^(7/2),x]

[Out]

(1 + x)^(5/2)/(5*(1 - x)^(5/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(1+x)^{3/2}}{(1-x)^{7/2}} \, dx &=\frac{(1+x)^{5/2}}{5 (1-x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0071806, size = 20, normalized size = 1. \[ \frac{(x+1)^{5/2}}{5 (1-x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^(3/2)/(1 - x)^(7/2),x]

[Out]

(1 + x)^(5/2)/(5*(1 - x)^(5/2))

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{5} \left ( 1+x \right ) ^{{\frac{5}{2}}} \left ( 1-x \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(3/2)/(1-x)^(7/2),x)

[Out]

1/5*(1+x)^(5/2)/(1-x)^(5/2)

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Maxima [B]  time = 1.04933, size = 127, normalized size = 6.35 \begin{align*} \frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1} + \frac{6 \, \sqrt{-x^{2} + 1}}{5 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{5 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{5 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(7/2),x, algorithm="maxima")

[Out]

(-x^2 + 1)^(3/2)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 6/5*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 1/5*sqrt(-x^2
+ 1)/(x^2 - 2*x + 1) - 1/5*sqrt(-x^2 + 1)/(x - 1)

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Fricas [B]  time = 1.55843, size = 130, normalized size = 6.5 \begin{align*} \frac{x^{3} - 3 \, x^{2} -{\left (x^{2} + 2 \, x + 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + 3 \, x - 1}{5 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(7/2),x, algorithm="fricas")

[Out]

1/5*(x^3 - 3*x^2 - (x^2 + 2*x + 1)*sqrt(x + 1)*sqrt(-x + 1) + 3*x - 1)/(x^3 - 3*x^2 + 3*x - 1)

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Sympy [B]  time = 23.4347, size = 88, normalized size = 4.4 \begin{align*} \begin{cases} - \frac{i \left (x + 1\right )^{\frac{5}{2}}}{5 \sqrt{x - 1} \left (x + 1\right )^{2} - 20 \sqrt{x - 1} \left (x + 1\right ) + 20 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\frac{\left (x + 1\right )^{\frac{5}{2}}}{5 \sqrt{1 - x} \left (x + 1\right )^{2} - 20 \sqrt{1 - x} \left (x + 1\right ) + 20 \sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**(3/2)/(1-x)**(7/2),x)

[Out]

Piecewise((-I*(x + 1)**(5/2)/(5*sqrt(x - 1)*(x + 1)**2 - 20*sqrt(x - 1)*(x + 1) + 20*sqrt(x - 1)), Abs(x + 1)/
2 > 1), ((x + 1)**(5/2)/(5*sqrt(1 - x)*(x + 1)**2 - 20*sqrt(1 - x)*(x + 1) + 20*sqrt(1 - x)), True))

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Giac [A]  time = 1.07877, size = 26, normalized size = 1.3 \begin{align*} -\frac{{\left (x + 1\right )}^{\frac{5}{2}} \sqrt{-x + 1}}{5 \,{\left (x - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^(3/2)/(1-x)^(7/2),x, algorithm="giac")

[Out]

-1/5*(x + 1)^(5/2)*sqrt(-x + 1)/(x - 1)^3